Nodal domains in open microwave systems

Nodal domains are studied both for real ${\psi }_R$ and imaginary part ${\psi }_I$ of the wave functions of an open microwave cavity and found to show the same behavior as wave functions in closed billiards. In addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase ${\phi }_g$ according to ${\psi }_R+i{\psi }_I=e^{i{\phi }_g}({\psi }_R^{\prime }+i{\psi }_I^{\prime })$. This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard.

Figure 1

(Color online) Real part ${\psi }_R$ (left-hand side), imaginary part ${\psi }_I$ (center), and modulus $\mid \psi {\mid }^2={\psi }_R^2+{\psi }_I^2$ of the wave function $\psi$ at $\nu =5.64\mathrm{GHz}$ corresponding to a Weyl number of $n_{\mathrm{Weyl}}=32.5$. Correlations between ${\psi }_R$ and ${\psi }_I$ have already been removed by a global phase, see text. Additionally the nodal points of the modulus and the nodal lines of real and imaginary parts are marked. Nodal points occur at crossings of nodal lines of ${\psi }_R$ with that of ${\psi }_I$.

Figure 2

Imaginary versus real parts of the wave function at $\nu =13.84\mathrm{GHz}$, for the directly measured wave function (left-hand side), and after uncorrelating them (right-hand side) by ${\psi }_R+i{\psi }_I=e^{-i{\phi }_{g,0}}({\psi }_R^{\prime }+i{\psi }_I^{\prime })$. ${\phi }_{g,0}$ is a global phase from the experiment acquired due to the antenna and leads.

Figure 3

(Color online) The number of nodal domains ${\nu }_n$ versus the Weyl number $n_{\mathrm{Weyl}}$ for the real (crosses) and imaginary (diamonds) parts. The (red) dashed line corresponds to the slope of 0.0624 predicted by the percolation model [4]. The (blue) solid lines are fits including boundary effects [2] with Eq. (6) with a linear slope of 0.059 and 0.060 for real and imaginary parts, respectively.

Figure 4

(Color online) The variance of the number of nodal domains ${\nu }_n$ versus the Weyl number $n_{\mathrm{Weyl}}$ for the real (blue boxes) and imaginary (red crosses) parts. The (blue) dashed line corresponds to the theoretical prediction.

Figure 5

(Color online) The distribution of normalized nodal domain areas $P_A(s∕s_{\min })$ is shown for the real (solid histogram) and the imaginary parts [(red) dotted histogram]. Additionally the expected algebraic decay with an exponent of $\tau =187∕91$ is shown as (blue) dashed line.

Figure 6

(Color online) (Upper part) Number of nodal domains ${\nu }_n$ as a function of a global phase ${\phi }_g$ for a wave function at $n_{\mathrm{Weyl}}\approx 223$. Marked are the phases ${\phi }_g$, for which the nodal domains are shown below, corresponding to ${\phi }_g∕\pi =0$, 0.225, 0.325, 0.475, 0.5, and 0.5525. Areas, where appearances and disappearances of nodal domains can be seen, are highlighted. The insets show the corresponding ${\psi }_I$ versus ${\psi }_R$ plots, see also Fig. 2.

Figure 7

Number of nodal domains ${\nu }_n$ as a function of a global phase ${\phi }_g$ for a wave function at $n_{\mathrm{Weyl}}\approx 229$.

Figure 8

(Color online) Correlation of the signed areas as a function of the global phase ${\phi }_g$ for three different wave functions at Weyl numbers $n_{\mathrm{Weyl}}\approx 229$, 222, and 223 with rigidities $\mid {\rho }^2\mid =0.04$, 0.48, and 0.81, respectively. The curve with the smallest curvature belong to $\mid {\rho }^2\mid =0.04$, and the one with the largest curvature belong to $\mid {\rho }^2\mid =0.81$. The dotted lines show the results of Eq. (12). The first and the last correlation correspond to the wave functions used to produce Figs. 6, 7.

Figure 9

(Color online) On the left-hand side the probability current density $\vec{j}$ is shown for the whole billiard at $13.84\mathrm{GHz}$. On the right-hand side the corresponding vorticity $\omega$ is plotted. The lower part of the figure shows a magnification of the region marked by squares in the upper figure. In the zoomed figures vortices and saddles are marked by circles and crosses. Open circles denote clockwise and filled circles denote counterclockwise vortices.

Figure 10

(Color online) The number of nodal domains ${\nu }_n$ versus the Weyl number $n_{\mathrm{Weyl}}$ for the vorticity $\omega$ (diamonds). The (red) dashed line correspond to the theoretical prediction ${\nu }_n=0.0624n$ of the percolation model [4]. The (blue) solid lines are fits including boundary effects [2] with Eq. (6) with a linear slope $a=0.161$.